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Ab initio study of the pseudorotation in 1,3-dioxolane
Journal of Molecular Structure 599 (2001) 271±278
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Ab initio study of the pseudorotation in 1,3-dioxolane q Jan Makarewicz a, Tae-Kyu Ha b,* a
b
Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, PL-60 780 PoznanÂ, Poland Laboratorium fuÈr Physikalische Chemie, Swiss Federal Institute of Technology, ETH Zentrum, CH 8092 ZuÈrich, Switzerland Received 14 August 2000; accepted 22 November 2000
Abstract Equilibrium molecular structures, vibrational wavenumbers, and the potential energy surface for the twisting and wagging large amplitude motions of the ring in 1,3-dioxolane have been calculated at the second-order Mùller±Plesset (MP2) level of perturbation theory using an extended basis set up to 6-311 1 G(3df,3p). The existence of four energy minima has been con®rmed by the calculations. Furthermore, a dynamical pseudorotation model has been derived from the calculated four local minima that are separated by low barriers, and the potential energy surface. The results of the present theoretical study have been critically compared with those of previous experimental studies as well as with previous lower level theoretical studies. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Ab initio study; 1,3-Dioxolane; Pseudorotation; Potential energy surface
1. Introduction A typical representative of saturated ®ve-membered ring molecules with ¯exible puckered conformations is 1,3-dioxolane (DOX). The ring puckering deformations of this molecule occurs due to a strong coupling of two out-of-plane large amplitude motions, the twisting and wagging motions of the ring. The dynamics of the ring puckering deformations can be conveniently described as a pseudorotation, i.e. a dynamic puckering displacement moving around the ring [1±4]. The amplitude r of the ring puckering and a phase angle f around the ring allow a simple q This paper is dedicated to Professor Alfred Bauder in appreciation of his signi®cant contributions to the ®eld of microwave spectroscopy. * Corresponding author. Tel.: 141-1-632-11-11; fax: 144-1-63210-21. E-mail address: [email protected] (T.-K. Ha).
description of the two-dimensional ring motion because these two modes are approximately separable. The pseudorotation of DOX can be well described by a one-dimensional model involving only the phase angle as a dynamical variable. The amplitude r and the remaining normal vibrations of the molecule can be treated as the motions adiabatically adjusted to the phase angle f . The potential energy function of the molecule varying along f plays the role of the onedimensional pseudorotation potential V f: This potential for DOX molecule exhibits very low potential barriers hindering the pseudorotational motion. For this reason, the conformations of DOX are ¯exional and dif®cult in the analysis based on the experimental data. The DOX molecule has been studied by a number of methods. It has been established that the ring is non-planar but its conformation is still not certain. The electron diffraction studies of Shen et al. [5]
0022-2860/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0022-286 0(01)00830-4
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J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278
Fig. 1. De®nition of the ring wagging and the twisting coordinates of DOX. The dihedral angle ,X2C4X1C2 de®nes the wagging coordinate a . The dihedral angle ,X2X1C4O3 which is equal to ,X2X1C5O1 de®nes the twisting coordinate b . The grey (white) circles represent the oxygen (carbon) nuclei.
have been interpreted in terms of a pseudorotation model with a two-fold potential energy function. Using a least-squares ®t of the experimental data they have found two symmetrically equivalent twisted conformations of this molecule as the energy minima separated by a low potential barrier estimated to be below 1.1 kcal mol 21. However, the electron diffraction data were little sensitive to the different conformers, so the results from these studies could not be considered as conclusive. The far-infrared spectra of DOX were interpreted [6,7] using the free pseudorotation model. However, this model could not explain the doublet splitting of the observed transitions between lower pseudorotation states. Such a splitting can be treated as a strong evidence for the low potential barriers in the potential V f: The microwave spectra of DOX have been observed for nine pseudorotation states and the splitting of the lowest energy level doublet has been precisely measured by Baron and Harris [8]. From this information, the pseudorotation potential V f with four equivalent minima has been determined. Each minimum corresponded to a half bent and half twisted ring con®guration. In a phenomenological model of pseudorotation considered in Ref. [9], the effective mass for this motion has not been calculated from a dynamical model but assumed to be independent of the angle f . In addition, this angle has not been strictly de®ned. As a consequence, the molecular structure, especially the equilibrium structure, corresponding to a given f could not be speci®ed. The spectra of DOX have been recently measured in the millimeter wave range and four sublevels due to
the pseudorotation potential with four-fold minima have been observed [9]. The structure and energetics of the DOX molecule have also been investigated using quantum chemistry ab initio methods. In the ®rst ab initio calculations by Cremer et al. [10,11] only a part of molecular geometry parameters were optimized since some of them were ®xed. The potential energy surface was not scanned in a wide range but only few stationary points were studied. In the optimization, rather small basis sets were applied and the electron correlation was not included. The results of the calculations strongly depended on the basis set employed. The twist conformation was obtained as a minimum by using 4-31G basis set whereas STO-3G basis yielded the envelope conformer as the stable one. In recent ab initio study of the DOX conformations [12], the full geometry optimization has been carried out but without including electron correlation. The twist form has been found as a preferred conformation, which was in disagreement with the molecular mechanics calculations [5] predicting the minimum envelope form. Ab inito predictions supported the conclusions from the electron diffraction studies but disagreed with more precise microwave spectroscopic results. To our knowledge, none ab initio work reproducing four local minima in the potential energy surface of DOX. The goal of this paper is to resolve inconsistency in the experimental and theoretical results on the structure and dynamics of the DOX molecule. We applied the ab initio method to construct the dynamical pseudorotation model of DOX. The electronic energy of this molecule has been calculated for scanned ring twisting and wagging coordinates and then the reliable two-dimensional potential energy surface has been obtained. The existence of four energy minima separated by low potential energy barriers has been revealed. The dynamical pseudorotation model, determined from the potential energy surface, allowed us to calculate the pseudorotation states. The information derived from the model has been used in the analysis of the experimental data. 2. Ab initio calculation of the potential energy surface The two-dimensional dynamical model of the DOX
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273
Table 1 Ê and angles in degrees) as calculated at the MP2(full)/6-311 1 G(2df,2p) level. The Parameters of the DOX ring geometry (bond lengths R in A numbering of the nuclei is given in accordance with Fig. 1 Parameter R(O1C2) R(O3C2) R(O1C5) R(O2C4) R(C4C5) , O1C2O3 , C2O1C5 , C2O3C4 , C4C5O1 , C5C4O3 Twist a Wag a
Experimental [5] 1.423
Experimental [14] 1.430 1.415
1.542 108.7 105.8
1.530 107.6 106.8
101.0
102.1
20.9 ±
19.2 ±
Ab initio, this work 1.414 1.400 1.420 1.428 1.524 106.9 106.8 102.7 103.8 102.6 13.0 11.6
Half of the dihedral angle ,O3C4C5O1.
molecule can be speci®ed by two coordinates, the ring wagging angle a and the ring twisting angle b . These angles characterize the bent and twisted con®gurations of the DOX ring and are de®ned in Fig. 1, where only the ring nuclei are shown. The dummy nuclei X1 and X2 were introduced to adopt these angles to the Z-matrix commonly used in ab initio programs. The remaining coordinates p fully specifying the structure of the DOX molecule were de®ned employing the standard valence coordinates, the bond lengths, bond angles and dihedral angles. They were treated as ¯exible geometry parameters, i.e. they were optimized for each ®xed point (a , b ) from the set of 50 points considered. These points were obtained by varying a and b from 0 to 258 in step of 58 and additional points were located near the potential energy minima, after calculation of the energy on the regular grid. The optimization was carried out using MP2 method (full treatment) and the basis set 6311 1 G(2df,2p). The ®nal energy was calculated at optimized structures applying the extended basis set 6-31111G(3df,3p). All computations of the electronic energy were carried out using the gaussian94 package programs [13]. The basis set 6-311 1 G(2df,2p) was chosen after test calculations which revealed that the optimized structure of the ring was very sensitive to the quality of the basis set. The optimization with the basis set 631G(d), often used in such type calculations, lead to the stable twisted ring structure with a; b 0; 20:3
and con®rmed earlier ab initio results [10±12]. However, this result appeared to be incorrect. A slightly larger basis set 6-31 1 G(d) with included diffuse orbitals on heavy atoms yielded a bent-twisted structure with a; b 7:6; 17:5: This structure was conserved under extension of the basis set, but the equilibrium angles (a , b ) changed signi®cantly. They achieved the values of (14.9, 6.1), and (11.6, 13.0) for the basis sets 6-311 1 G(d,p) and 6311 1 G(2df,2p), respectively. This shows that the diffuse orbitals and the polarized f orbitals on heavy atoms are necessary to achieve an accurate optimized ring structure. The calculated equilibrium structure parameters of DOX are compared in Table 1 to the experimental ones, derived from electron diffraction experiment. Some of them are averaged, so they cannot be precisely compared to the calculated values. The agreement of the ab initio and experimental [14] parameters is reasonable except for the twisting angle, which seems to be overestimated in experimental models. The wagging angle has not been determined since the model of Ref. [6] assumed only the stable twist conformations. As a consequence, the experimental bond length R(O1C2) also seems to be overestimated due to a drawback of the model. The quality of the ab initio results can be veri®ed by comparing the calculated and measured fundamental vibrational frequencies. Barker et al. [15] recorded the infrared and Raman spectra and assigned all fundamental vibrational modes, except for the low
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J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278
Table 2 Comparison of the observed and calculated normal vibration frequencies (cm 21) of dioxolan. The ab initio results refer to the planar and bent (in equilibrium) molecule Vibration Sym. CH stretch (C4,C5) Sym. CH stretch (C2) CH2 scissors (C2) Sym. CH2 scissors (C4,C5) Sym. CH2 wag (C4,C5) Ring stretch Ring stretch Ring in plane bend Ring breathing Antisym. CH stretch (C4,C5) CH2 twist (C4,C5) Antisym. CH2 twist (C4,C5) Antisym. CH2 rock (C4,C5) Antisym. CH stretch (C2) Antisym. CH stretch (C4,C5) Sym. CH2 twist (C4,C5) Sym. CH2 rock (C4,C5) CH2 rock (C2) Sym. CH stretch (C4,C5) Antisym. CH2 scissors C4,C5) CH2 wag (C2) Antisym. CH2 wag (C4,C5) Ring stretch Ring stretch Ring in-plane bend a
Infrared a A1 symmetry 2889 2857 1509 1480 1361 1087 1030 939 ± A2 symmetry ± 1251 1208 ± B1 symmetry 2998 2964 1286 921 723 B2 symmetry 2889 1480 1397 1327 1158 961 680
Raman a
Ab initio planar
Ab initio bent
2894 2852 1509 1481 1352 1088 1038 939 658
3112 3073 1584 1564 1398 1203 1018 967 751
3113 3035 1559 1542 1370 1200 1005 957 739
2972 1246 1210 1009
3147 1262 1231 1162
3192 1295 1223 1165
± 2972 ± ± 725
3172 3126 1253 1168 865
3206 3170 1256 1133 887
2894 1481 1397 1329 ± 962 671
3104 1549 1456 1389 1137 974 741
3078 1524 1436 1370 1087 983 668
Ref. [15].
frequency ring twisting and wagging modes. They classi®ed these modes according the symmetry of the planar DOX, since the expected degree of puckering in the DOX ring was assumed to be small. Under this assumption, the molecule has a two-fold symmetry axis (the z-axis) and two symmetry planes, that of the ring (yz) and that (xz) perpendicular to the ring plane. These symmetry elements de®ne the C2v point group. To conserve the classi®cation of the normal modes according to the symmetry classes of the C2v point group [15] we calculated their frequencies for the planar DOX con®guration, which is a saddle point but not a minimum on the potential energy surface. Just two vibrational modes, the ring twisting (n b ) and the wagging (n a ) were characterized by negative frequencies, so they were not consid-
ered in this case. The analogous vibrational frequencies calculated for the bent ring at the potential energy minimum were all positive. Table 2 compares the observed [15] and ab initio vibrational frequencies calculated at the MP2(full)/6-311 1 G(2df,2p) level of theory. The present theoretical results con®rm the assignment of Barker et al. [15], except for the A1 ring breathing and ring bent modes, which are reversed in our assignment. It should be also mentioned that two of the CH2 rocking mode assignments (B1) agree poorly with the calculations, probably badly assigned in the infrared and Raman spectra. Naturally, the large amplitude anharmonic modes n a and n b could not be accurately determined in the framework of the harmonic normal mode scheme. For this purpose, the two-dimensional potential energy
J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278
275
surface V a; b of DOX as a function of the a and b dynamical variables was obtained by ®tting the polynomial of these variables to the calculated ab initio energy values. Similarly, each optimized geometry parameter was represented as a function p(a , b ). The contour plot of V a; b is shown in Fig. 2. Note, that the apparent symmetry of V a; b can be described by the following two generating symmetry operations a: (a , b ) ! (2a , b ) and b: (a , b ) ! (a , 2b ) so they constitute the symmetry group isomorphic to C2v point group because a2 b2 1: So this group properly describes the symmetry of the vibrational modes. 3. Pseudorotation model Fig. 2. Contour plot of the wagging±twisting potential energy surface V a; b of DOX, calculated at the MP2 (full)/6311 1 G(3df,3p)//MP2(full)/6-311 1 G(2df,2p) level of the ab initio theory.
In order to model the pseudorotational motion in DOX, new `polar' coordinates f and r were further introduced through the relations
a Ar sin f b r cos f
Fig. 3. Contour plot of the ab initio potential energy surface of DOX, expressed in polar coordinates f and r .
1
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J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278
Table 3 Fourier coef®cients ck, and barrier heights Vb(0) and Vb(90) at 0 and 908 of the potential energy functions V1 f; V2 f and VMW f: All quantities are in cm 21, except for the pseudorotation (f wagging (a ), and twisting (b ) angles at the minimum given in degrees ck
V1
V2
VMW a
c0 c2 c4 c6 c8 Vb(0) Vb(90) f min a min b min
83.20 2 12.61 2 33.09 2 4.40 2 0.94 83.20 49.18 44.8 13.7 10.0
52.20 14.34 2 31.07 2 4.40 2 0.94 55.4 72.3 38.5 15.2 9.7
0.0 2 5.1 2 20.0 ± ± 45.3 35.0 46.8 ± ±
a
Ref. [8].
The coef®cient A de®nes the form of the ellipse when the phase angle f varies from 0 to 2p at the ®xed radius r . For A 0:8 this ellipse is close to the minimum energy path along the angle f , connecting four equivalent local minima on the potential energy surface. This surface expressed in polar coordinates, V r; f; is shown in Fig. 3. The minimum energy path was determined by minimizing V r; f at the ®xed values of f from the range (0±2p ). As a result, the radius r optf which plays the role of the puckering amplitude, was determined as a slowly varying function of the angle f The minimum energy along the path can be treated as the one-dimensional pseudorotation potential function V1 f V ropt f; f: The remaining geometry parameters of DOX were obtained also as functions of the leading variable f As a consequence, the dynamical model of the pseudorotation was speci®ed and used to determine numerically the kinetic part of the Hamiltonian ^ f according to the method proposed by Meyer H [16]. The pseudorotation wavefunctions c r(f ) and energy levels E r r 0; 1; 2; ¼ were calculated as ^ f; using the eigenfunctions of the operator H program described in Ref. [17]. The pseudorotation potential function V1 f was represented by the Fourier expansion X V1 f c0 1 ck 1 2 cos kf 2 k
whose coef®cients, nonzero only for even k, are collected in Table 3. The potential V1 f) and the
Fig. 4. Pseudopotential energy functions the DOX molecule: the potential V1 f determined from ab initio calculations (a), the potential V2(f ) including the zero-point vibrational energy (b) and the empirical potential VMW(f ) ®tted in Ref. [8] to the microwave spectra (c).
J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278 Table 4 Pseudorotation energy levels of DOX calculated from the potential energy functions V1, V2, and VMW. The energy values are given in cm 21 relative to the ground state energy level r 0 r
V1
V2
VMW a
1 2 3 4 5 6 7 8 9 10
1.47 5.71 8.34 40.51 49.24 59.95 81.60 89.90 125.99 127.94
1.43 5.90 8.50 41.16 52.02 57.25 81.99 88.39 126.37 126.63
2.16 4.43 8.47 28.65 40.18 42.77 66.66 69.79 103.62 103.84
a
Ref. [8].
®rst pseudorotation energy levels of DOX calculated from V1 f are illustrated in Fig. 4a. This potential with four-fold minima is qualitatively similar to the potential VMW f determined from the microwave spectra [8] and yields a similar energy level pattern of the lowest states. The above presented ¯exible model takes into account only the static part of the interaction of the pseudorotational motion with the remaining modes of higher frequencies. Such an interaction is described as an adjustment of the molecular structure to the pseudorotation of the ring. However, this model can be corrected by including the contribution DV of the 3N 2 5 high frequency vibrational modes to the potential V1 f: This contribution is simply a zeropoint vibrational energy excluding the energy of the pseudorotation: DV f
1 2
"
3N 25 X n1
vn f
3
The correction DV was calculated only for the stationary points, namely, for the minimum, and two saddle points at f 0 and 908. The following values were obtained: DV fmin 20758:1 cm21 ; DV 0 20727:5 cm21 and DV 90 20781:5 cm21 : These three values are suf®cient to approximate DV f by the Fourier expansion DV f=cm21 ù 230:60 1 27:05 1 2 cos 2f 1 1:82 1 2 cos 4f;
4
277
where the coef®cient c0 is chosen such that DV fmin 0: The modi®ed potential energy function V2 f V1 f 1 DV f is compared to the V1 f and VMW f potentials in Figs. 4a±c and their properties are summarized in Table 3. The ab initio values of f min cannot be directly compared to the corresponding empirical value, because the phase angle f in the empirical model plays the role of the unspeci®ed phenomenological variable. For this reason, the wagging and twisting angles cannot be determined from the empirical value f min. The correction DV f slightly changes the effective structure of the ring by increasing the twisting angle at the potential minimum. However, DV f signi®cantly lowers the potential barriers at the twisted con®gurations f 0 and ^ 1808 and increases the barriers at the envelope con®gurations f ^908: As a consequence, the heights of the twist and the envelope barriers are reversed. In the potential energy function VMW f determined experimentally, the twist barrier was established to be 10 cm 21 higher in energy than the envelope barrier. This means, that the vibrational zero-point correction DV changes the barriers in V1 f in a right direction, but too strongly. As a consequence, the calculated splitting of the lowest energy doublet is smaller than the observed one. The pseudorotation energy levels calculated from the ab initio potentials V1 f and V2 f are compared in Table 4 to the corresponding levels calculated from the experimental potential VMW f The splitting pattern of the lowest quartet of the pseudorotation energy levels is reproduced reasonably. However, the higher energy levels are shifted by about 10 cm 21 relative to the energy levels determined from VMW f because of too high potential energy barriers in the ab initio potentials. The reported in Table 4 pseudorotation energy levels are very similar to the corresponding energy levels determined from the two-dimensional potential energy surface V r; f; so they are not reported. The energy of 301 cm 21 calculated from V r; f for the ®rst excited radial state, with the node in the radial coordinate r is in excellent agreement with the frequency of 302 cm 21 observed in the Raman spectra [6].
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J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278
4. Conclusions The two-dimensional potential energy surface describing the coupled wagging and twisting ring motions in DOX was determined for the ®rst time from the ab initio calculations. The electron correlation was taken into account at the MP2 level. The electronic energy of the molecule was calculated on a discrete set of the wagging and twisting variables (a , b ). At each considered point, the remaining molecular geometry parameters were optimized using the basis set 6-311 1 G(2df,2p) and the ®nal single-point energy was calculated using the larger basis set 6311 1 G(3df,3p). This large basis set appeared to be well saturated because the obtained potential energy surface differed insigni®cantly from that calculated with the 6-311 1 G(2df,2p) basis set. The latter basis was employed in the calculation of the fundamental vibrational frequencies. The results of calculations con®rmed most of the assigned infrared and Raman bands [15] and corrected assignment of two bands corresponding to ring vibrations. From the twodimensional potential energy surface, the frequency of the radial puckering mode was calculated to be 301 cm 21, which agrees with the observed Raman frequency of 302 cm 21. The dynamical pseudorotation model was determined from the calculated potential V a; b and optimized molecular structure as function of the variables (a , b ). The obtained one-dimensional pseudorotation potential V1 f and the potential V2 f including the zero-point energy of the highfrequency vibrations revealed four equivalent local minima separated by low potential barriers at the twisted and bent (envelope) con®gurations. This ®nding strongly supports the results from the analysis of the microwave spectra which allowed the determination of the empirical pseudorotation potential VMW f: Although the barriers of the ab initio potentials appeared to be overestimated, the pseudorotation energy level patterns of the ®rst quartet are similar for all discussed potentials. This fact may help in a direct observation of the splitting. The analysis of the molecular structure clearly indicates that the equilibrium twisting angle determined from the electron diffraction studies [5] is too large. This is a consequence of the incorrect model of pseudorotation with assumed potential energy
function exhibiting two equivalent local minima corresponding to the twisted ring. This assumption was based on early ab initio calculations [10,11] which yielded unreliable structures. This work shows, that a reliable ab initio description of ®ve-membered ring molecules needs a large basis set including f polarized orbitals to obtain accurate optimized structures and an adequate dynamical model of the pseudorotation. Acknowledgements The authors thank Professor W. Caminati for bringing our attention to inconsistent interpretation of the experimental data for DOX. We thank also Dr B. Brupbacher-Gatehouse for reading and improving the manuscript. References [1] J.E. Kilpatrick, K.S. Pitzer, R. Spitzer, J. Am. Chem. Soc. 69 (1947) 2483. [2] H.L. Strauss, Ann. Rev. Phys. Chem. 301 (1983) 301. [3] J. Laane, in: J. Durig (Ed.), Vibrational Spectra and Structure, vol. I, Dekker, New York, 1972, p. 25. [4] J. Laane, J. Pure Appl. Chem. 59 (1987) 1307. [5] O. Shen, T.L. Mathers, T. Raeker, R.L. Hildebrandt, J. Am. Chem. Soc. 108 (1986) 6888. [6] J.R. Durig, D.W. Wertz, J. Chem. Phys. 49 (1968) 675. [7] J.A. Greenhouse, H.L. Strauss, J. Chem. Phys. 50 (1969) 124. [8] P.A. Baron, D.O. Harris, J. Mol. Spectrosc. 49 (1974) 70. [9] Results of W. Caminati, A. Maris, cited in the paper by G. Maccaferri, H. Dreizler, W. Caminati, J. Mol. Spectrosc., 196 (1999) 338. [10] D. Cremer, J.A. Pople, J. Am. Chem. Soc. 97 (1975) 1358. [11] D. Cremer, Isr. J. Chem. 23 (1983) 72. [12] A. Skancke, L. Vilkov, Acta Chem. Scand. A42 (1988) 717. [13] M.J. Frish, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. AlLaham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts,. R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. HeadGordon, C. Gonzalez, J.A. Pople, gaussian 94, Revision C.3 (Gaussian, Inc., Pittsburgh PA, 1995). [14] Results of K. Hedberg cited in Ref. [5]. [15] S.A. Barker, E.J. Bourne, R.M. Pinkard, D.H. Whiffen, J. Am. Soc. 161 (1959) 802. [16] R. Meyer, J. Mol. Spectrosc. 76 (1979) 266. [17] J. Makarewicz, J. Mol. Spectrosc. 176 (1996) 169.
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Ab initio study of the pseudorotation in 1,3-dioxolane q Jan Makarewicz a, Tae-Kyu Ha b,* a
b
Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, PL-60 780 PoznanÂ, Poland Laboratorium fuÈr Physikalische Chemie, Swiss Federal Institute of Technology, ETH Zentrum, CH 8092 ZuÈrich, Switzerland Received 14 August 2000; accepted 22 November 2000
Abstract Equilibrium molecular structures, vibrational wavenumbers, and the potential energy surface for the twisting and wagging large amplitude motions of the ring in 1,3-dioxolane have been calculated at the second-order Mùller±Plesset (MP2) level of perturbation theory using an extended basis set up to 6-311 1 G(3df,3p). The existence of four energy minima has been con®rmed by the calculations. Furthermore, a dynamical pseudorotation model has been derived from the calculated four local minima that are separated by low barriers, and the potential energy surface. The results of the present theoretical study have been critically compared with those of previous experimental studies as well as with previous lower level theoretical studies. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Ab initio study; 1,3-Dioxolane; Pseudorotation; Potential energy surface
1. Introduction A typical representative of saturated ®ve-membered ring molecules with ¯exible puckered conformations is 1,3-dioxolane (DOX). The ring puckering deformations of this molecule occurs due to a strong coupling of two out-of-plane large amplitude motions, the twisting and wagging motions of the ring. The dynamics of the ring puckering deformations can be conveniently described as a pseudorotation, i.e. a dynamic puckering displacement moving around the ring [1±4]. The amplitude r of the ring puckering and a phase angle f around the ring allow a simple q This paper is dedicated to Professor Alfred Bauder in appreciation of his signi®cant contributions to the ®eld of microwave spectroscopy. * Corresponding author. Tel.: 141-1-632-11-11; fax: 144-1-63210-21. E-mail address: [email protected] (T.-K. Ha).
description of the two-dimensional ring motion because these two modes are approximately separable. The pseudorotation of DOX can be well described by a one-dimensional model involving only the phase angle as a dynamical variable. The amplitude r and the remaining normal vibrations of the molecule can be treated as the motions adiabatically adjusted to the phase angle f . The potential energy function of the molecule varying along f plays the role of the onedimensional pseudorotation potential V f: This potential for DOX molecule exhibits very low potential barriers hindering the pseudorotational motion. For this reason, the conformations of DOX are ¯exional and dif®cult in the analysis based on the experimental data. The DOX molecule has been studied by a number of methods. It has been established that the ring is non-planar but its conformation is still not certain. The electron diffraction studies of Shen et al. [5]
0022-2860/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0022-286 0(01)00830-4
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J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278
Fig. 1. De®nition of the ring wagging and the twisting coordinates of DOX. The dihedral angle ,X2C4X1C2 de®nes the wagging coordinate a . The dihedral angle ,X2X1C4O3 which is equal to ,X2X1C5O1 de®nes the twisting coordinate b . The grey (white) circles represent the oxygen (carbon) nuclei.
have been interpreted in terms of a pseudorotation model with a two-fold potential energy function. Using a least-squares ®t of the experimental data they have found two symmetrically equivalent twisted conformations of this molecule as the energy minima separated by a low potential barrier estimated to be below 1.1 kcal mol 21. However, the electron diffraction data were little sensitive to the different conformers, so the results from these studies could not be considered as conclusive. The far-infrared spectra of DOX were interpreted [6,7] using the free pseudorotation model. However, this model could not explain the doublet splitting of the observed transitions between lower pseudorotation states. Such a splitting can be treated as a strong evidence for the low potential barriers in the potential V f: The microwave spectra of DOX have been observed for nine pseudorotation states and the splitting of the lowest energy level doublet has been precisely measured by Baron and Harris [8]. From this information, the pseudorotation potential V f with four equivalent minima has been determined. Each minimum corresponded to a half bent and half twisted ring con®guration. In a phenomenological model of pseudorotation considered in Ref. [9], the effective mass for this motion has not been calculated from a dynamical model but assumed to be independent of the angle f . In addition, this angle has not been strictly de®ned. As a consequence, the molecular structure, especially the equilibrium structure, corresponding to a given f could not be speci®ed. The spectra of DOX have been recently measured in the millimeter wave range and four sublevels due to
the pseudorotation potential with four-fold minima have been observed [9]. The structure and energetics of the DOX molecule have also been investigated using quantum chemistry ab initio methods. In the ®rst ab initio calculations by Cremer et al. [10,11] only a part of molecular geometry parameters were optimized since some of them were ®xed. The potential energy surface was not scanned in a wide range but only few stationary points were studied. In the optimization, rather small basis sets were applied and the electron correlation was not included. The results of the calculations strongly depended on the basis set employed. The twist conformation was obtained as a minimum by using 4-31G basis set whereas STO-3G basis yielded the envelope conformer as the stable one. In recent ab initio study of the DOX conformations [12], the full geometry optimization has been carried out but without including electron correlation. The twist form has been found as a preferred conformation, which was in disagreement with the molecular mechanics calculations [5] predicting the minimum envelope form. Ab inito predictions supported the conclusions from the electron diffraction studies but disagreed with more precise microwave spectroscopic results. To our knowledge, none ab initio work reproducing four local minima in the potential energy surface of DOX. The goal of this paper is to resolve inconsistency in the experimental and theoretical results on the structure and dynamics of the DOX molecule. We applied the ab initio method to construct the dynamical pseudorotation model of DOX. The electronic energy of this molecule has been calculated for scanned ring twisting and wagging coordinates and then the reliable two-dimensional potential energy surface has been obtained. The existence of four energy minima separated by low potential energy barriers has been revealed. The dynamical pseudorotation model, determined from the potential energy surface, allowed us to calculate the pseudorotation states. The information derived from the model has been used in the analysis of the experimental data. 2. Ab initio calculation of the potential energy surface The two-dimensional dynamical model of the DOX
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273
Table 1 Ê and angles in degrees) as calculated at the MP2(full)/6-311 1 G(2df,2p) level. The Parameters of the DOX ring geometry (bond lengths R in A numbering of the nuclei is given in accordance with Fig. 1 Parameter R(O1C2) R(O3C2) R(O1C5) R(O2C4) R(C4C5) , O1C2O3 , C2O1C5 , C2O3C4 , C4C5O1 , C5C4O3 Twist a Wag a
Experimental [5] 1.423
Experimental [14] 1.430 1.415
1.542 108.7 105.8
1.530 107.6 106.8
101.0
102.1
20.9 ±
19.2 ±
Ab initio, this work 1.414 1.400 1.420 1.428 1.524 106.9 106.8 102.7 103.8 102.6 13.0 11.6
Half of the dihedral angle ,O3C4C5O1.
molecule can be speci®ed by two coordinates, the ring wagging angle a and the ring twisting angle b . These angles characterize the bent and twisted con®gurations of the DOX ring and are de®ned in Fig. 1, where only the ring nuclei are shown. The dummy nuclei X1 and X2 were introduced to adopt these angles to the Z-matrix commonly used in ab initio programs. The remaining coordinates p fully specifying the structure of the DOX molecule were de®ned employing the standard valence coordinates, the bond lengths, bond angles and dihedral angles. They were treated as ¯exible geometry parameters, i.e. they were optimized for each ®xed point (a , b ) from the set of 50 points considered. These points were obtained by varying a and b from 0 to 258 in step of 58 and additional points were located near the potential energy minima, after calculation of the energy on the regular grid. The optimization was carried out using MP2 method (full treatment) and the basis set 6311 1 G(2df,2p). The ®nal energy was calculated at optimized structures applying the extended basis set 6-31111G(3df,3p). All computations of the electronic energy were carried out using the gaussian94 package programs [13]. The basis set 6-311 1 G(2df,2p) was chosen after test calculations which revealed that the optimized structure of the ring was very sensitive to the quality of the basis set. The optimization with the basis set 631G(d), often used in such type calculations, lead to the stable twisted ring structure with a; b 0; 20:3
and con®rmed earlier ab initio results [10±12]. However, this result appeared to be incorrect. A slightly larger basis set 6-31 1 G(d) with included diffuse orbitals on heavy atoms yielded a bent-twisted structure with a; b 7:6; 17:5: This structure was conserved under extension of the basis set, but the equilibrium angles (a , b ) changed signi®cantly. They achieved the values of (14.9, 6.1), and (11.6, 13.0) for the basis sets 6-311 1 G(d,p) and 6311 1 G(2df,2p), respectively. This shows that the diffuse orbitals and the polarized f orbitals on heavy atoms are necessary to achieve an accurate optimized ring structure. The calculated equilibrium structure parameters of DOX are compared in Table 1 to the experimental ones, derived from electron diffraction experiment. Some of them are averaged, so they cannot be precisely compared to the calculated values. The agreement of the ab initio and experimental [14] parameters is reasonable except for the twisting angle, which seems to be overestimated in experimental models. The wagging angle has not been determined since the model of Ref. [6] assumed only the stable twist conformations. As a consequence, the experimental bond length R(O1C2) also seems to be overestimated due to a drawback of the model. The quality of the ab initio results can be veri®ed by comparing the calculated and measured fundamental vibrational frequencies. Barker et al. [15] recorded the infrared and Raman spectra and assigned all fundamental vibrational modes, except for the low
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Table 2 Comparison of the observed and calculated normal vibration frequencies (cm 21) of dioxolan. The ab initio results refer to the planar and bent (in equilibrium) molecule Vibration Sym. CH stretch (C4,C5) Sym. CH stretch (C2) CH2 scissors (C2) Sym. CH2 scissors (C4,C5) Sym. CH2 wag (C4,C5) Ring stretch Ring stretch Ring in plane bend Ring breathing Antisym. CH stretch (C4,C5) CH2 twist (C4,C5) Antisym. CH2 twist (C4,C5) Antisym. CH2 rock (C4,C5) Antisym. CH stretch (C2) Antisym. CH stretch (C4,C5) Sym. CH2 twist (C4,C5) Sym. CH2 rock (C4,C5) CH2 rock (C2) Sym. CH stretch (C4,C5) Antisym. CH2 scissors C4,C5) CH2 wag (C2) Antisym. CH2 wag (C4,C5) Ring stretch Ring stretch Ring in-plane bend a
Infrared a A1 symmetry 2889 2857 1509 1480 1361 1087 1030 939 ± A2 symmetry ± 1251 1208 ± B1 symmetry 2998 2964 1286 921 723 B2 symmetry 2889 1480 1397 1327 1158 961 680
Raman a
Ab initio planar
Ab initio bent
2894 2852 1509 1481 1352 1088 1038 939 658
3112 3073 1584 1564 1398 1203 1018 967 751
3113 3035 1559 1542 1370 1200 1005 957 739
2972 1246 1210 1009
3147 1262 1231 1162
3192 1295 1223 1165
± 2972 ± ± 725
3172 3126 1253 1168 865
3206 3170 1256 1133 887
2894 1481 1397 1329 ± 962 671
3104 1549 1456 1389 1137 974 741
3078 1524 1436 1370 1087 983 668
Ref. [15].
frequency ring twisting and wagging modes. They classi®ed these modes according the symmetry of the planar DOX, since the expected degree of puckering in the DOX ring was assumed to be small. Under this assumption, the molecule has a two-fold symmetry axis (the z-axis) and two symmetry planes, that of the ring (yz) and that (xz) perpendicular to the ring plane. These symmetry elements de®ne the C2v point group. To conserve the classi®cation of the normal modes according to the symmetry classes of the C2v point group [15] we calculated their frequencies for the planar DOX con®guration, which is a saddle point but not a minimum on the potential energy surface. Just two vibrational modes, the ring twisting (n b ) and the wagging (n a ) were characterized by negative frequencies, so they were not consid-
ered in this case. The analogous vibrational frequencies calculated for the bent ring at the potential energy minimum were all positive. Table 2 compares the observed [15] and ab initio vibrational frequencies calculated at the MP2(full)/6-311 1 G(2df,2p) level of theory. The present theoretical results con®rm the assignment of Barker et al. [15], except for the A1 ring breathing and ring bent modes, which are reversed in our assignment. It should be also mentioned that two of the CH2 rocking mode assignments (B1) agree poorly with the calculations, probably badly assigned in the infrared and Raman spectra. Naturally, the large amplitude anharmonic modes n a and n b could not be accurately determined in the framework of the harmonic normal mode scheme. For this purpose, the two-dimensional potential energy
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275
surface V a; b of DOX as a function of the a and b dynamical variables was obtained by ®tting the polynomial of these variables to the calculated ab initio energy values. Similarly, each optimized geometry parameter was represented as a function p(a , b ). The contour plot of V a; b is shown in Fig. 2. Note, that the apparent symmetry of V a; b can be described by the following two generating symmetry operations a: (a , b ) ! (2a , b ) and b: (a , b ) ! (a , 2b ) so they constitute the symmetry group isomorphic to C2v point group because a2 b2 1: So this group properly describes the symmetry of the vibrational modes. 3. Pseudorotation model Fig. 2. Contour plot of the wagging±twisting potential energy surface V a; b of DOX, calculated at the MP2 (full)/6311 1 G(3df,3p)//MP2(full)/6-311 1 G(2df,2p) level of the ab initio theory.
In order to model the pseudorotational motion in DOX, new `polar' coordinates f and r were further introduced through the relations
a Ar sin f b r cos f
Fig. 3. Contour plot of the ab initio potential energy surface of DOX, expressed in polar coordinates f and r .
1
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Table 3 Fourier coef®cients ck, and barrier heights Vb(0) and Vb(90) at 0 and 908 of the potential energy functions V1 f; V2 f and VMW f: All quantities are in cm 21, except for the pseudorotation (f wagging (a ), and twisting (b ) angles at the minimum given in degrees ck
V1
V2
VMW a
c0 c2 c4 c6 c8 Vb(0) Vb(90) f min a min b min
83.20 2 12.61 2 33.09 2 4.40 2 0.94 83.20 49.18 44.8 13.7 10.0
52.20 14.34 2 31.07 2 4.40 2 0.94 55.4 72.3 38.5 15.2 9.7
0.0 2 5.1 2 20.0 ± ± 45.3 35.0 46.8 ± ±
a
Ref. [8].
The coef®cient A de®nes the form of the ellipse when the phase angle f varies from 0 to 2p at the ®xed radius r . For A 0:8 this ellipse is close to the minimum energy path along the angle f , connecting four equivalent local minima on the potential energy surface. This surface expressed in polar coordinates, V r; f; is shown in Fig. 3. The minimum energy path was determined by minimizing V r; f at the ®xed values of f from the range (0±2p ). As a result, the radius r optf which plays the role of the puckering amplitude, was determined as a slowly varying function of the angle f The minimum energy along the path can be treated as the one-dimensional pseudorotation potential function V1 f V ropt f; f: The remaining geometry parameters of DOX were obtained also as functions of the leading variable f As a consequence, the dynamical model of the pseudorotation was speci®ed and used to determine numerically the kinetic part of the Hamiltonian ^ f according to the method proposed by Meyer H [16]. The pseudorotation wavefunctions c r(f ) and energy levels E r r 0; 1; 2; ¼ were calculated as ^ f; using the eigenfunctions of the operator H program described in Ref. [17]. The pseudorotation potential function V1 f was represented by the Fourier expansion X V1 f c0 1 ck 1 2 cos kf 2 k
whose coef®cients, nonzero only for even k, are collected in Table 3. The potential V1 f) and the
Fig. 4. Pseudopotential energy functions the DOX molecule: the potential V1 f determined from ab initio calculations (a), the potential V2(f ) including the zero-point vibrational energy (b) and the empirical potential VMW(f ) ®tted in Ref. [8] to the microwave spectra (c).
J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278 Table 4 Pseudorotation energy levels of DOX calculated from the potential energy functions V1, V2, and VMW. The energy values are given in cm 21 relative to the ground state energy level r 0 r
V1
V2
VMW a
1 2 3 4 5 6 7 8 9 10
1.47 5.71 8.34 40.51 49.24 59.95 81.60 89.90 125.99 127.94
1.43 5.90 8.50 41.16 52.02 57.25 81.99 88.39 126.37 126.63
2.16 4.43 8.47 28.65 40.18 42.77 66.66 69.79 103.62 103.84
a
Ref. [8].
®rst pseudorotation energy levels of DOX calculated from V1 f are illustrated in Fig. 4a. This potential with four-fold minima is qualitatively similar to the potential VMW f determined from the microwave spectra [8] and yields a similar energy level pattern of the lowest states. The above presented ¯exible model takes into account only the static part of the interaction of the pseudorotational motion with the remaining modes of higher frequencies. Such an interaction is described as an adjustment of the molecular structure to the pseudorotation of the ring. However, this model can be corrected by including the contribution DV of the 3N 2 5 high frequency vibrational modes to the potential V1 f: This contribution is simply a zeropoint vibrational energy excluding the energy of the pseudorotation: DV f
1 2
"
3N 25 X n1
vn f
3
The correction DV was calculated only for the stationary points, namely, for the minimum, and two saddle points at f 0 and 908. The following values were obtained: DV fmin 20758:1 cm21 ; DV 0 20727:5 cm21 and DV 90 20781:5 cm21 : These three values are suf®cient to approximate DV f by the Fourier expansion DV f=cm21 ù 230:60 1 27:05 1 2 cos 2f 1 1:82 1 2 cos 4f;
4
277
where the coef®cient c0 is chosen such that DV fmin 0: The modi®ed potential energy function V2 f V1 f 1 DV f is compared to the V1 f and VMW f potentials in Figs. 4a±c and their properties are summarized in Table 3. The ab initio values of f min cannot be directly compared to the corresponding empirical value, because the phase angle f in the empirical model plays the role of the unspeci®ed phenomenological variable. For this reason, the wagging and twisting angles cannot be determined from the empirical value f min. The correction DV f slightly changes the effective structure of the ring by increasing the twisting angle at the potential minimum. However, DV f signi®cantly lowers the potential barriers at the twisted con®gurations f 0 and ^ 1808 and increases the barriers at the envelope con®gurations f ^908: As a consequence, the heights of the twist and the envelope barriers are reversed. In the potential energy function VMW f determined experimentally, the twist barrier was established to be 10 cm 21 higher in energy than the envelope barrier. This means, that the vibrational zero-point correction DV changes the barriers in V1 f in a right direction, but too strongly. As a consequence, the calculated splitting of the lowest energy doublet is smaller than the observed one. The pseudorotation energy levels calculated from the ab initio potentials V1 f and V2 f are compared in Table 4 to the corresponding levels calculated from the experimental potential VMW f The splitting pattern of the lowest quartet of the pseudorotation energy levels is reproduced reasonably. However, the higher energy levels are shifted by about 10 cm 21 relative to the energy levels determined from VMW f because of too high potential energy barriers in the ab initio potentials. The reported in Table 4 pseudorotation energy levels are very similar to the corresponding energy levels determined from the two-dimensional potential energy surface V r; f; so they are not reported. The energy of 301 cm 21 calculated from V r; f for the ®rst excited radial state, with the node in the radial coordinate r is in excellent agreement with the frequency of 302 cm 21 observed in the Raman spectra [6].
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J. Makarewicz, T.-K. Ha / Journal of Molecular Structure 599 (2001) 271±278
4. Conclusions The two-dimensional potential energy surface describing the coupled wagging and twisting ring motions in DOX was determined for the ®rst time from the ab initio calculations. The electron correlation was taken into account at the MP2 level. The electronic energy of the molecule was calculated on a discrete set of the wagging and twisting variables (a , b ). At each considered point, the remaining molecular geometry parameters were optimized using the basis set 6-311 1 G(2df,2p) and the ®nal single-point energy was calculated using the larger basis set 6311 1 G(3df,3p). This large basis set appeared to be well saturated because the obtained potential energy surface differed insigni®cantly from that calculated with the 6-311 1 G(2df,2p) basis set. The latter basis was employed in the calculation of the fundamental vibrational frequencies. The results of calculations con®rmed most of the assigned infrared and Raman bands [15] and corrected assignment of two bands corresponding to ring vibrations. From the twodimensional potential energy surface, the frequency of the radial puckering mode was calculated to be 301 cm 21, which agrees with the observed Raman frequency of 302 cm 21. The dynamical pseudorotation model was determined from the calculated potential V a; b and optimized molecular structure as function of the variables (a , b ). The obtained one-dimensional pseudorotation potential V1 f and the potential V2 f including the zero-point energy of the highfrequency vibrations revealed four equivalent local minima separated by low potential barriers at the twisted and bent (envelope) con®gurations. This ®nding strongly supports the results from the analysis of the microwave spectra which allowed the determination of the empirical pseudorotation potential VMW f: Although the barriers of the ab initio potentials appeared to be overestimated, the pseudorotation energy level patterns of the ®rst quartet are similar for all discussed potentials. This fact may help in a direct observation of the splitting. The analysis of the molecular structure clearly indicates that the equilibrium twisting angle determined from the electron diffraction studies [5] is too large. This is a consequence of the incorrect model of pseudorotation with assumed potential energy
function exhibiting two equivalent local minima corresponding to the twisted ring. This assumption was based on early ab initio calculations [10,11] which yielded unreliable structures. This work shows, that a reliable ab initio description of ®ve-membered ring molecules needs a large basis set including f polarized orbitals to obtain accurate optimized structures and an adequate dynamical model of the pseudorotation. Acknowledgements The authors thank Professor W. Caminati for bringing our attention to inconsistent interpretation of the experimental data for DOX. We thank also Dr B. Brupbacher-Gatehouse for reading and improving the manuscript. References [1] J.E. Kilpatrick, K.S. Pitzer, R. Spitzer, J. Am. Chem. Soc. 69 (1947) 2483. [2] H.L. Strauss, Ann. Rev. Phys. Chem. 301 (1983) 301. [3] J. Laane, in: J. Durig (Ed.), Vibrational Spectra and Structure, vol. I, Dekker, New York, 1972, p. 25. [4] J. Laane, J. Pure Appl. Chem. 59 (1987) 1307. [5] O. Shen, T.L. Mathers, T. Raeker, R.L. Hildebrandt, J. Am. Chem. Soc. 108 (1986) 6888. [6] J.R. Durig, D.W. Wertz, J. Chem. Phys. 49 (1968) 675. [7] J.A. Greenhouse, H.L. Strauss, J. Chem. Phys. 50 (1969) 124. [8] P.A. Baron, D.O. Harris, J. Mol. Spectrosc. 49 (1974) 70. [9] Results of W. Caminati, A. Maris, cited in the paper by G. Maccaferri, H. Dreizler, W. Caminati, J. Mol. Spectrosc., 196 (1999) 338. [10] D. Cremer, J.A. Pople, J. Am. Chem. Soc. 97 (1975) 1358. [11] D. Cremer, Isr. J. Chem. 23 (1983) 72. [12] A. Skancke, L. Vilkov, Acta Chem. Scand. A42 (1988) 717. [13] M.J. Frish, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. AlLaham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts,. R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. HeadGordon, C. Gonzalez, J.A. Pople, gaussian 94, Revision C.3 (Gaussian, Inc., Pittsburgh PA, 1995). [14] Results of K. Hedberg cited in Ref. [5]. [15] S.A. Barker, E.J. Bourne, R.M. Pinkard, D.H. Whiffen, J. Am. Soc. 161 (1959) 802. [16] R. Meyer, J. Mol. Spectrosc. 76 (1979) 266. [17] J. Makarewicz, J. Mol. Spectrosc. 176 (1996) 169.